Related papers: A Fractional Lie Group Method For Anomalous Diffus…
The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of…
The object of this paper is to present a computable solution of a fractional partial differential equation associated with a Riemann-Liouville derivative of fractional order as the time-derivative and Riesz-Feller fractional derivative as…
Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of…
Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those diffusions, the…
In this paper, the Lie symmetry analysis is proposed for a space-time convection-diffusion fractional differential equations with the Riemann-Liouville derivative by (2+1) independent variables and one dependent variable. We find a…
A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time…
In this article, we consider discrete schemes for a fractional diffusion equation involving a tempered fractional derivative in time. We present a semi-discrete scheme by using the local discontinuous Galerkin (LDG) discretization in the…
The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical…
Fractional kinetic equations employ non-integer calculus to model anomalous relaxation and diffusion in many systems. While this approach is well explored, it so far failed to describe an important class of transport in disordered systems.…
A fractional diffusion equation with advection term is rigorously derived from a kinetic transport model with a linear turning operator, featuring a fat-tailed equilibrium distribution and a small directional bias due to a given vector…
In this paper, a nonlinear system of fractional ordinary differential equations with multiple scales in time is investigated. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the…
In view of the role of reaction equations in physical problems, the authors derive the explicit solution of a fractional reaction equation of general character, that unifies and extends earlier results. Further, an alternative shorter…
The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of…
Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the…
We consider difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters $\alpha$, $\beta$ and $\gamma$. By the method of energy inequalities, for the…
A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order…
In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
In this paper we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…